Differential Geometry and its Applications最新文献

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.

We give a complete classification of isoparametric hypersurfaces in a product space $M_{{\kappa }_1}^2\times M_{{\kappa }_2}^2$ of 2-dimensional space forms for ${\kappa }_i\in \{-1,0,1\}$ with ${\kappa }_1\ne {\kappa }_2$. In fact we prove that any isoparametric hypersurface in such a space has constant product angle function, which enables us to remove the condition of constant principal curvatures from the classification obtained recently by J.B.M. dos Santos and J.P. dos Santos.

The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of absolutely continuous curves coincide when their domains are bounded closed intervals. As an application, a universal existence and regularity theorem for gradient flow is obtained in the Finsler setting. Secondly, we study absolutely continuous curves in Wasserstein spaces over Finsler manifolds and establish the Lisini structure theorem in this setting, which characterize the nature of absolutely continuous curves in Wasserstein spaces in terms of dynamical transference plans concentrated on absolutely continuous curves in base Finsler manifolds. Besides, a close relation between continuity equations and absolutely continuous curves in Wasserstein spaces is founded. Last but not least, we also consider nonsmooth “Finsler-like” spaces, in which case most of the aforementioned results remain valid. Various model examples are constructed in this paper, which point out genuine differences between the asymmetric and symmetric settings.

We show that a compact almost-Kähler four-manifold $(M,g,\omega )$ with harmonic self-dual Weyl curvature and constant scalar curvature is Kähler if $c_1\cdot \left[\omega \right]\ge 0$. We also prove an integral curvature inequality for compact almost-Kähler four-manifolds with harmonic self-dual Weyl curvature.

Weak almost contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak (2022), allowed a new look at the theory of contact manifolds. This paper studies the curvature and topology of new structures of this type, called the weak nearly cosymplectic structure and weak nearly Kähler structure. We find conditions under which weak nearly cosymplectic manifolds become Riemannian products and characterize 5-dimensional weak nearly cosymplectic manifolds. Our theorems generalize results by H. Endo (2005) and A. Nicola–G. Dileo–I. Yudin (2018) to the context of weak almost contact geometry.

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension ≥3.

Previously we finish the establishment of the transversality of the general part of the Vafa-Witten moduli spaces, in this paper, we deal with the rest, i.e., the reduced part. We consider Vafa-Witten equation on closed, oriented and smooth Riemann 4-manifolds with $C\equiv 0$, and construct perturbation to establish the transversality of the perturbed equation. We show that for a generic choice of the perturbation terms, the moduli space of solutions to the perturbed reduced Vafa-Witten equation for the structure group $SU\left(2\right)$ or $SO\left(3\right)$ on a closed 4-manifold is a smooth manifold of dimension zero. Finally we prove that for two generic orientation-preserving parameters, the corresponding moduli spaces are cobordant, and the method can also be applied to the general part.

Let $P:=U(p+q)/U\left(p\right)\times U\left(q\right)$ be the complex Grassmann manifold and $F:T^{1,0}P\to [0,+\infty )$ be an arbitrary $U(p+q)$-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical $U(p+q)$-invariant Kähler metric on $P$. In particular on the complex projective space ${\mathrm{CP}}^n=U(n+1)/U\left(n\right)\times U\left(1\right)$, there exists no $U(n+1)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on $P$ which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the $U(p+q)$-invariant Kähler metrics on $P$, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

This study is focused on the investigation of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) spacetime. Recently, Poyraz (2022) [51], [52] established some basic inequalities involving various curvature invariants of screen homothetic lightlike hypersurfaces of GRW spacetimes, like k-scalar curvature and k-Ricci curvature. In this work, we consider other basic curvature invariants, namely the scalar curvature and δ-Casorati curvatures, and derive new inequalities for such hypersurfaces of a GRW spacetime. We also find the conditions for which the equality cases in these inequalities hold and give some applications in Lorentzian geometry.